Strain-stiffening Effect Research Articles

Development of keratin-based fibers fabricated by interfacial polyelectrolyte complexation for suture applications

Interfacial Polyelectrolyte Complexation (IPC) is a convenient way to produce composite, micro-scale fibers. In this paper, we report the successful development of novel keratin-based IPC fibers and also demonstrate the feasibility of using these fibers as sutures through a proof-of-concept in vivo study. Two composite fibers were produced: chitosan-keratin (CK) and keratin-keratin (KK). These fibers were evaluated for their physico-chemical, mechanical and biochemical properties. In the dry state, the CK fiber had a greater Young's modulus of about 2 GPa while the KK fiber registered a longer strain-at-break of about 100 % due to the strain-stiffening effect. Notably, the keratins were found to assemble into amyloids within the composite fibers based on Congo red staining and Wide-Angle X-Ray Scattering. Functionally, both fibers were malleable could be weaved, braided and knotted. When used as sutures to close incisional wounds in mice over 21 days, these fibers were found to elicit minimal host tissue response and were partially degraded over the duration. Interestingly, the KK fiber evoked a lower extent of immune cell response and fibrous capsule encapsulation that was comparable to commercial, non-absorbable Dafilon® sutures. This work demonstrated the possibility of producing keratin-based IPC fibers which may find practicality as medical sutures.

Electroactive differential growth and delayed instability in accelerated healing tissues

Guided by experiments contrasting electrically accelerated recovery with natural healing, this study formulates a model to investigate the importance of electroactive differential growth and morphological changes in tissue repair. It underscores the clinical potential of leveraging electroactive differential growth for improved healing outcomes. The study reveals that voltage stimulation significantly enhances the healing and growth of biological tissues, accelerating the regeneration process across various growth modalities and steering towards isotropic growth conditions that do not favor any specific growth pathways. Enhancing the electroelastic coupling parameters improves the efficacy of bioelectric devices, initiating contraction and fortification of biological tissues in alignment with the electric field. This process facilitates swift cell migration and proliferation, as well as oriented growth of tissue. In instances of strain stiffening at elevated strains, the extreme critical growth ratio aligns with the predictions of neo-Hookean models. Conversely, for tissues experiencing strain stiffening under moderate to very low strain conditions, the strain stiffening effect substantially delays the onset of electroelastic growth instability, ultimately producing a smooth, hyperelastic surface devoid of any unstable morphologies. Our investigation, grounded in nonlinear electroelastic field and perturbation theories, explores how electric fields influence differential growth and instability in biological tissues. We examine the interactions among dimensionless voltage, internal pressure, electroelastic coupling, radius ratio, and strain stiffening, revealing their effects on promoting growth and delaying instability. This framework offers insights into the mechanisms behind electroactive growth and its instabilities, contributing valuable knowledge to the tissue healing.

Approximate analytical solutions of nonlinear vibration and bifurcation of dielectric elastomer balloon

Dielectric elastomers (DE), a novel class of actuators, are increasingly employed in fields such as soft robotics and vibration control. Most existing studies investigating the dynamic characteristics of DE structures neglect the viscoelasticity and strain-stiffening effects of materials, with only a few offering approximate analytical solutions. This paper establishes the governing equation for the dynamics of a DE balloon, utilizing the principle of virtual work and considering material viscoelasticity and strain-stiffening effects through linear damping and the Gent hyperelastic model, respectively. The nonlinear governing equations are simplified using the Chebyshev polynomial approximation. The multiple scales method is applied to derive approximate solutions for the structure's free vibration under static voltage excitation and parametrically excited vibration under harmonic voltage excitation. The accuracy of these approximate analytical solutions is validated by comparison with numerical solutions. Additionally, the paper analyzes stability and bifurcation phenomena through time-domain response curves, amplitude-frequency curves, and phase trajectories. Furthermore, the influence of the stretching limit, voltage, and damping on the nonlinear vibration of the balloon structure is studied, providing a valuable theoretical foundation for the future design and utilization of dielectric elastomers.

Propagation of the Fundamental Lamb Modes in Strain Stiffened Hard-Magnetic Soft Plates

Abstract This study focuses on investigating hard-magnetic soft materials, characterized by magnetoactive polymers containing magnetically polarized particles as fillers. The research utilizes the Gent model of hyperelasticity to analyze the propagation of Lamb waves in a magnetically induced deformed compressible plate. In this investigation, we explore both finite deformations and incremental wave propagation in nonlinear hard-magnetic soft materials. The main objective is to formulate the elastic tensor and relevant wave equations within the framework of Lagrangian space. To assess the dispersion characteristics of the guided wave, the study introduces and discusses an extension of the semi-analytical finite element (SAFE) method. Using this numerical approach, the research further examines the effects of magnetic flux densities and its orientation with respect to wave propagation direction on the dispersion characteristics of the fundamental Lamb modes. The study starts by examining the limiting case of the neo-Hookean material model to explain such inherent dependencies. These dependencies are then further emphasized by including the strain-stiffening effect that the Gent material model describes. The research findings reveal the presence of a threshold applied magnetic flux, beyond which the Gent-type material may undergo a snap-through instability, resulting in changes in the dispersion characteristics of the fundamental symmetric Lamb mode.

Voltage-controlled non-axisymmetric vibrations of soft electro-active tubes with strain-stiffening effect

Material properties of soft electro-active (SEA) structures are significantly sensitive to external electro-mechanical biasing fields (such as pre-stretch and electric stimuli), which generate remarkable knock-on effects on their dynamic characteristics. In this work, we analyze the electrostatically tunable non-axisymmetric vibrations of an incompressible SEA cylindrical tube under the combination of a radially applied electric voltage and an axial pre-stretch. Following the theory of nonlinear electro-elasticity and the associated linearized theory for superimposed perturbations, we derive the nonlinear static response of the SEA tube to the inhomogeneous biasing fields for the Gent ideal dielectric model. Using the State Space Method, we efficiently obtain the frequency equations for voltage-controlled small-amplitude three-dimensional non-axisymmetric vibrations, covering a wide range of behaviors, from the purely radial breathing mode to torsional modes, axisymmetric longitudinal modes, and prismatic diffuse modes. We also perform an exhaustive numerical analysis to validate the proposed approach compared with the conventional displacement method, as well as to elucidate the influences of the applied voltage, axial pre-stretch, and strain-stiffening effect on the nonlinear static response and vibration behaviors of the SEA tube. The present study clearly indicates that manipulating electro-mechanical biasing fields is a feasible way to tune the small-amplitude vibration characteristics of an SEA tube. The results should benefit experimental work on, and design of, voltage-controlled resonant devices made of SEA tubes.

Analysis of the Mullins effect in buckling instability of double-network hydrogel beams under swelling equilibrium

Due to their outstanding physical and mechanical properties, double-network (DN) hydrogels have gained a most attention among various synthetic tough hydrogels. The Mullins effect and buckling instability are two frequent phenomena that may occur simultaneously in slender DN hydrogel structures as high load-bearing candidates. The swelling/deswelling degree of the DN hydrogel affects the coupling phenomena. It is essential to understanding the interplay between these behaviors. In this work, a physically based damage constitutive model for a DN polymer under water-diffusion equilibrium and cyclic loadings is developed. This theory of coupled swelling and load-induced deformations show good capability in capturing the Mullins effect of swollen DN hydrogels and the corresponding swelling ratio. The model parameters are determined by fitting experimental data of a freely swollen DN hydrogel under cyclic compression. Based on the constitutive model and determined parameters, the buckling instability of DN hydrogel beams is studied via the analytical formula of incremental modulus. The stability diagrams of the DN hydrogel beams under virgin loading and reloading are presented. The influences of stress softening, strain stiffening and chemical potential on buckling conditions for compressive stress and slenderness ratio are thoroughly analyzed. It demonstrates that the stress softening is dramatically against the stability of the beam, but the strain-stiffening effect would conversely help widen the stable range. Besides, it is found that a DN gel beam immersed in a sufficient low chemical potential environment has better buckling stability. These theoretical results are valuable in the preparation and structural design of DN hydrogels for repeated use purpose.

The mechanics of embedded fiber networks

Fiber networks underlie the mechanical behavior of a wide range of natural and engineered materials. Interestingly, these networks are often embedded within amorphous matrices rather than appearing in isolation. However, despite their frequent occurrence as embedded rather than isolated networks, few prior studies have focused on investigating the role of embedding on the emergent mechanical behavior of these systems. To address this, we adopt a mortar-type embedding approach within the finite element framework and perform simulations to systematically fill this knowledge gap. Within this study, we focus on soft tissues as an exemplary class of materials where embedded fiber networks are essential to mechanical function. Specifically, we investigate the role of embedding on the strain energy distribution within the networks across the bending, stretching, torsional, and shear fiber-level loading modes. Therein, we specifically focus on semi-flexible fiber networks. In addition to revealing the role of embedding on the networks themselves, we also investigate how the networks affect the mechanics of the matrix material. Together, we find that embedding fundamentally alters the mechanics of semi-flexible fiber networks and the surrounding matrices. Most importantly, we find that embedding semi-flexible fiber networks leads to strain-stiffening and negative Poynting effect of the resulting composite material. Furthermore, semi-flexible fiber networks induce stress heterogeneity in their host material and increase its resistance to compression. Overall, our work improves our fundamental understanding of an important class of materials. By making our implementation openly available, we also hope to help others learn more about embedded semi-flexible fiber networks in the context of materials other than soft tissues.

Strain stiffening retards growth instability in residually stressed biological tissues

Soft biological tissues often exhibit notable strain stiffening under increasing stretch, and this can have significant effects on tissue growth and morphological development, such as causing symmetry breaking in growing airways and leading to mucosal folding and airway hyperresponsiveness. To investigate the role of strain stiffening and the multifactorial control in growth and remodeling, we consider a growing tubular structure with strain-stiffening effects caused by increased and tightened collagen. In addition, we employ the nonlinear hyperelastic Gent model and initial stress symmetry theory to include the coupling effects of differential growth and initial residual stress. Results show that for strain stiffening that takes place at higher strain (Jm>21), the maximum critical growth ratio matches that obtained using neo-Hookean model calculations. Meanwhile, for biological tissues that exhibit strain stiffening under moderate strain conditions (0.46<Jm<21), the strain-stiffening effect delays significantly the onset of growth instability. When strain stiffening takes place at very low strains (Jm<0.46), stiff biological tissues can prevent growth instability, resulting in a smooth hyperelastic cylindrical tubular structure, and the epithelial tissue remains stable at all growth stages without forming any unstable morphology. Our results suggest that strain stiffening can induce retardation instability during biological growth and remodeling, but airway remodeling can incorporate this effect by increasing wall stiffness and reducing obstruction. This highlights the importance of considering the impact of strain stiffening on biological growth and remodeling, which can inform the development of effective clinical interventions for chronic inflammatory airway diseases.

Resilience study of wheat protein networks with large amplitude oscillatory shear rheology

This study investigates the effect of standard laboratory production steps on the network properties of vital gluten. Gluten experiences high mechanical input during production from wheat dough and subsequent processing, which influences the network characteristics. The production of vital gluten in laboratory scale ensures full control on all processing parameters and the possibility to obtain the protein using the mildest washing, drying and milling techniques. Large Amplitude Oscillatory Shear (LAOS) tests were performed on wheat dough, intermediate materials during gluten production and the final product vital gluten at frequencies of 1–15 rad/s and strains from 0.01 to 100%. The elastic and viscous behavior of the material are described by analyzing the fundamental response and the higher harmonics, based on Lissajous-Bowditch curves. The intra-cycle behavior revealed that vital gluten only exhibits network stiffening, whereas dough shows both strain stiffening and shear thinning effects. This suggests that starch induces the shear thinning effects in dough.Comparison of rehydrated and fresh gluten obtained from the same starting material reveals minor alteration of the protein secondary structures during the production process by means of Fourier-transform infrared spectroscopy (FTIR). A pronounced difference in water uptake accounts for modified rheological network properties.

Propagation of fundamental Lamb modes along the non-principal axes of strain-stiffened soft compressible plates: A numerical investigation.

This paper numerically investigates the propagation of elastic plate waves along the non-principal directions in a prestretched compressible material described by the Gent model of hyperelasticity. We formulate the elastic tensor and the underlying wave equations in the Lagrangian space by employing the theory of nonlinear elasticity together with the linearized incremental equations. An extension of the Semi-Analytical Finite Element (SAFE) method is discussed for computing the dispersion characteristics of the two fundamental guided wave modes. The predictive capabilities of the numerical framework are established using the previously published data for a weakly nonlinear as well as hyperelastic material models. Using the numerical framework, we then bring out the effects of applied prestretch, orientation of the propagation direction, and material parameters on the dispersion characteristics of the fundamental Lamb modes. A limiting case of the neo-Hookean material model is first considered for elucidating such implicit dependencies, which are further highlighted by considering the strain-stiffening effect captured through the Gent material model. Our results indicate the existence of a threshold prestretch for which the Gent-type material can encounter a snap-through instability; leading to the change in the dispersion characteristics of the fundamental symmetric Lamb mode.

Bionic artificial penile Tunica albuginea

Bionic artificial penile Tunica albuginea

A Semi-Analytical Finite Element Framework for Lamb Waves in Soft Compressible Plates Considering Strain Stiffening Effect

This paper investigates the elastic wave propagation through soft materials that are being subjected to finite deformations. The nonlinear elastic and linearized incremental theories have been exploited to formulate governing wave equations and elastic moduli in Lagrangian space. Semi-analytical finite element (SAFE) method, a numerical approach has been formulated for computing dispersive relations of guided waves in compressible hyper-elastic plates. This framework requires finite element discretization of the cross section of the waveguide and harmonic exponential function assumes the motion along the wave propagation direction. Here, explicit phase velocity results have been shown for soft materials with a prominent stiffening effect by employing the Gent model, and these results are analyzed for elastic wave propagation through compressible materials. It has been noticed that Lamb waves have a strong dependence on the frequency-thickness product, prestretch, and direction of wave propagation. Moreover, with the strain stiffening effect, the dependence becomes stronger, especially for fundamental symmetric and anti-symmetric modes. The numerical results display that at certain prestretch the Gent material encounter snap-through instability resulting from geometrical and material nonlinearities. The influence of material properties like Gent constant and direction of wave propagation on snap-through instability has been discussed. The proposed SAFE framework reveals that finite deformations can affect elastic wave propagation through stiffness and compressibility.

Exact large strain analysis for the Poynting effect of freely twisted thin-walled tubes made of highly elastic soft materials

Toward handling nonlinear coupling effects at large elastic deformations, a decoupled form of the multi-axial elastic potential for highly elastic soft materials is first constructed based on Hencky’s logarithmic strain, and this new potential is then used to obtain analytic results for the uniaxial extension and compression of a bar, the equi-biaxial extension of a plate, the plane-strain extension of a strip and, in particular, the large torsion of a thin-walled tube with free ends. Novelties are included in the following three respects: (a) Experimental data for responses of the uniaxial extension, equi-biaxial extension (uniaxial compression) and plane-strain extension (simple shear) may be accurately matched up to large strain with strain-stiffening effects; (b) exact responses for the Poynting effect of freely twisted thin-walled tubes may be derived for changes in the axial length and the wall thickness as well as the averaged radius; and (c) the coupling nonlinear complexities involved in all such responses may be worked out in a decoupled sense.

A new micro–macro transition for hyperelastic materials

Many classic hyperelastic models fail to predict the stress responses of soft materials in complex loading conditions with parameters calibrated through one simple test. To address this fundamental issue, we propose a new micro–macro transition for the microstructural hyperelasticity modeling, which is further integrated into the full network framework. With a Gaussian chain distribution, this new mapping scheme yields an explicit one-parameter hyperelastic model in terms of principal stretches. This new linear model achieves remarkable success in capturing the stress responses in multi-axial deformation modes for soft materials with an absence of strain stiffening effect, which is beyond the capability of the widely used neo-Hookean model. A new two-parameter hyperelastic model is further developed by combining the new micro–macro transition and non-Gaussian Langevin chain distribution. Compared with other two-parameter hyperelastic models based on Langevin statistics, such as the eight-chain model, affine full network model, and equilibrated microsphere model, our new model exhibits greatly improved predictive ability for complex loading types. The new model is also implemented for finite element analysis, which shows the ability to capture the responses of soft materials with heterogeneous strain distribution. In all cases, the parameters in our models can be determined through the data of uniaxial loading tests, along with the behaviors in other loading modes being well forecast, which is a challenge for various other existing hyperelastic models. This novel micro–macro transition is shown to properly capture the inherent correlations between different deformation modes, which may advance fundamentally modeling hyperelasticity and other constitutive behaviors for soft materials.

Alleviation of Residual Vibrations in Hard-Magnetic Soft Actuators Using a Command-Shaping Scheme.

Hard-magnetic soft materials belong to a class of the highly deformable magneto-active elastomer family of smart materials and provide a promising technology for flexible electronics, soft robots, and functional metamaterials. When hard-magnetic soft actuators are driven by a multiple-step input signal (Heaviside magnetic field signal), the residual oscillations exhibited by the actuator about equilibrium positions may limit their performance and accuracy in practical applications. This work aims at developing a command-shaping scheme for alleviating residual vibrations in a magnetically driven planar hard-magnetic soft actuator. The control scheme is based on the balance of magnetic and elastic forces at a critical point in an oscillation cycle. The equation governing the dynamics of the actuator is devised using the Euler–Lagrange equation. The constitutive behaviour of the hard-magnetic soft material is modeled using the Gent model of hyperelasticity, which accounts for the strain-stiffening effects. The dynamic response of the actuator under a step input signal is obtained by numerically solving the devised dynamic governing equation using MATLAB ODE solver. To demonstrate the applicability of the developed command-shaping scheme, a thorough investigation showing the effect of various parameters such as material damping, the sequence of desired equilibrium positions, and polymer chain extensibility on the performance of the proposed scheme is performed. The designed control scheme is found to be effective in controlling the motion of the hard-magnetic soft actuator at any desired equilibrium position. The present study can find its potential application in the design and development of an open-loop controller for hard-magnetic soft actuators.

Nonlinear Vibration and Stability of a Dielectric Elastomer Balloon Based on a Strain-Stiffening Model

Limiting chain extensibility is a characteristic that plays a vital role in the stretching of highly elastic materials. The Gent model has been widely used to capture this behaviour, as it performs very well in fitting stress-stretch data in simple tension, and involves two material parameters only. Recently, Anssari-Benam and Bucchi (Int. J. Non. Linear. Mech. 128:103626, 2021) introduced a different form of generalised neo-Hookean model, focusing on the molecular structure of elastomers, and showed that their model encompasses all ranges of deformations, performing better than the Gent model in many respects, also with only two parameters. Here we investigate the nonlinear vibration and stability of a dielectric elastomer balloon modelled by that strain energy function. We derive the deformation field in spherical coordinates and the governing equations by the Euler-Lagrange method, assuming that the balloon retains its spherical symmetry as it inflates. We consider in turn that the balloon is under two types of voltages, a pure DC voltage and an AC voltage superimposed on a DC voltage. We analyse the dynamic response of the balloon and identify the influential parameters in the model. We find that the molecular structure of the material, as tracked by the number of segments in a single chain, can control the instability and the pull-in/snap-through critical voltage, as well as chaos and quasi-periodicity. The main result is that balloons made of materials exhibiting early strain-stiffening effects are more stable and less prone to generate chaotic nonlinear vibrations than when made of softer materials, such as those modelled by the neo-Hookean strain-energy density function.

An alternative form of energy density demonstrating the severe strain-stiffening in thin spherical and cylindrical shells

The present article investigates an elastic instability phenomenon for internally pressurized spherical thin balloons and thin cylindrical tubes composed of incompressible hyperelastic material. A mathematical model is formulated by proposing a new strain energy density function. In the family of limited elastic materials, many material models exhibit strain-stiffening. However, they fail to predict severe strain-stiffening in a moderate range of deformations in the stress-strain relations. The proposed energy function contains three material parameters and shows substantially improved stain stiffening properties than the limited elastic material models. The model is further applied to explore the elastic instability phenomenon in spherical and cylindrical shells. The findings are compared with other existing models and validated with experimental results. The model shows better agreement with experimental results and exhibits a substantial strain-stiffening effect than the current models.

Functional mechanical attributes of natural and synthetic gel-based scaffolds in tissue engineering: strain-stiffening effects on apparent elastic modulus and compressive toughness

The accurate identification and determination of elastic modulus and toughness, as well as other functional mechanical attributes of artificial tissues, are of paramount importance in several fields of tissue science, tissue engineering and technology, since biomechanical and biophysical behavior is strongly linked to biological features of the medical implants and tissue-engineering scaffolds. When soft or ultra-soft materials are investigated, a relevant dispersion of elastic modulus values can be achieved, due to the strain-stiffening effects, inducing a typical non-linear behavior of these materials, as a function of strain-range. In this short communication, the Apparent elastic modulus strain-range dependence is estimated from a segmentation of the strain stiffening curve, and the related compressive toughness is investigated and discussed, based on experimental evidence, for 6 different kinds of gels, used for artificial tissue fabrication; experimental results are compared to mechanical properties of native human tissues.

Nonlinear Free and Forced Vibrations of a Hyperelastic Micro/Nanobeam Considering Strain Stiffening Effect.

In recent years, the static and dynamic response of micro/nanobeams made of hyperelasticity materials received great attention. In the majority of studies in this area, the strain-stiffing effect that plays a major role in many hyperelastic materials has not been investigated deeply. Moreover, the influence of the size effect and large rotation for such a beam that is important for the large deformation was not addressed. This paper attempts to explore the free and forced vibrations of a micro/nanobeam made of a hyperelastic material incorporating strain-stiffening, size effect, and moderate rotation. The beam is modelled based on the Euler–Bernoulli beam theory, and strains are obtained via an extended von Kármán theory. Boundary conditions and governing equations are derived by way of Hamilton’s principle. The multiple scales method is applied to obtain the frequency response equation, and Hamilton’s technique is utilized to obtain the free undamped nonlinear frequency. The influence of important system parameters such as the stiffening parameter, damping coefficient, length of the beam, length-scale parameter, and forcing amplitude on the frequency response, force response, and nonlinear frequency is analyzed. Results show that the hyperelastic microbeam shows a nonlinear hardening behavior, which this type of nonlinearity gets stronger by increasing the strain-stiffening effect. Conversely, as the strain-stiffening effect is decreased, the nonlinear frequency is decreased accordingly. The evidence from this study suggests that incorporating strain-stiffening in hyperelastic beams could improve their vibrational performance. The model proposed in this paper is mathematically simple and can be utilized for other kinds of micro/nanobeams with different boundary conditions.

Inversion and perversion in twist incompatible isotropic tubes

Tubular and rod-like structures in nature can experience twist due to non-torque loadings such as inflation and extension. Interestingly, the twist can change its direction and even vanish along the loading path. These counter-intuitive phenomena are referred to as inversion and perversion, respectively, and stem from the anisotropic structure of the tubes. In this work we show that a similar response can be achieved with bi-layer tubes comprising two isotropic layers that are twisted in different directions and adhered together. We refer to these tubes as geometrically incompatible. By employing the Gent model, we find that the inversion and perversion effects stem from the changes in the torsional stiffness of the layers as a result of inflation or extension. We derive relations for the torsional stiffness in terms of the geometry, the stiffness, the lock-up stretch which accounts for the strain-stiffening effect, and the deformation. We illustrate that different thickness and stiffness ratios between the layers lead to a variety of induced twist responses. To better understand the influence of the geometric and the material properties of the layers on the overall twist response, we map the combinations of thickness and stiffness ratios that lead to inversion and perversion and demonstrate the richness of these phenomena. The findings and insights from this work are of value to a wide range of applications, including the design of soft rotational robots, artificial muscles, and soft torsional actuators.